Followed by an entry where the Trudinger inequality had been discussed we now consider an important variant of it known as the Moser-Trudinger inequality.
Let us remind the Trudinger inequality
Theorem (Trudinger). Let
be a bounded domain and
with
.
Then there exist universal constants
,
such that
.
The Trudinger inequality has lots of application. For application to the prescribed Gauss curvature equation, one requires a particular value for the best constant
. In connection with his work on the Gauss curvature equation, J. Moser [here] sharpended the above result of Trungdier as follows
Theorem (Moser). Let
be a bounded domain and
with
.
Then there exist sharp constants
,
given by

such that
.
The constant
is sharp in the sense that for all
there is a sequence of functions
satisfying

but the integral

grow without bound.
For general compact closed manifold
the constant on the right hand side of the Moser-Trudinger inequality depends on the metric
. Working on a sphere
with a canonical metric allows us to control the constants.
Theorem (Moser). There is a universal constant
such that for all
with

and

we have
.
Observe that
.
In the same way as we introduce in the entry concerning the Trudinger inequality one can show
Corollary. For

one has
![\displaystyle \log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_2}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clog+%5Coverline%5Cint_%7B%7B%5Cmathbb%7BS%7D%5E2%7D%7D+%7B%7Be%5E%7B2u%7D%7Dd%7Bv_%7B%7Bg_c%7D%7D%7D%7D+%5Cleqslant+%5Cleft%5B+%7B%5Cfrac%7B1%7D%7B%7B4%5Cpi+%7D%7D%5Cint_%7B%7B%5Cmathbb%7BS%7D%5E2%7D%7D+%7B%7B%7B%5Cleft%7C+%7B%5Cnabla+u%7D+%5Cright%7C%7D%5E2%7Dd%7Bv_%7B%7Bg_c%7D%7D%7D%7D+%2B+2%5Coverline+u+%7D+%5Cright%5D+%2B+%7BC_2%7D&bg=ffffff&fg=333333&s=0&c=20201002)
for all
.
Obviously,
since
. It turns out to determine the best constant
. This had been done by Onofri known as the Onofri inequality [here].
Theorem (Onofri).Let
then we have

with the equality iff
.
The proof of the Onofri inequality relies on a result due to Aubin
Theorem (Aubin). For all
there exists a constant
such that
![\displaystyle\log \overline\int_{{\mathbb{S}^2}} {{e^{2u}}d{v_{{g_c}}}} \leqslant \left[ {\left( {\frac{1}{2} + \varepsilon } \right)\frac{1}{{4\pi }}\int_{{\mathbb{S}^2}} {{{\left| {\nabla u} \right|}^2}d{v_{{g_c}}}} + 2\overline u } \right] + {C_\varepsilon }](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Clog+%5Coverline%5Cint_%7B%7B%5Cmathbb%7BS%7D%5E2%7D%7D+%7B%7Be%5E%7B2u%7D%7Dd%7Bv_%7B%7Bg_c%7D%7D%7D%7D+%5Cleqslant+%5Cleft%5B+%7B%5Cleft%28+%7B%5Cfrac%7B1%7D%7B2%7D+%2B+%5Cvarepsilon+%7D+%5Cright%29%5Cfrac%7B1%7D%7B%7B4%5Cpi+%7D%7D%5Cint_%7B%7B%5Cmathbb%7BS%7D%5E2%7D%7D+%7B%7B%7B%5Cleft%7C+%7B%5Cnabla+u%7D+%5Cright%7C%7D%5E2%7Dd%7Bv_%7B%7Bg_c%7D%7D%7D%7D+%2B+2%5Coverline+u+%7D+%5Cright%5D+%2B+%7BC_%5Cvarepsilon+%7D&bg=ffffff&fg=333333&s=0&c=20201002)
for any
belonging to the following class
.
Source: S-Y.A. Chang, Non-linear elliptic equations in conformal geometry, EMS, 2004.